Capelli's Identity - Relations With Representation Theory

Relations With Representation Theory

Consider the following slightly more general context. Suppose that n and m are two integers and x_ij for i = 1,...,n,j = 1,...,m, be commuting variables. Redefine E_ij by almost the same formula:

with the only difference that summation index a ranges from 1 to m. One can easily see that such operators satisfy the commutation relations:

Here denotes the commutator . These are the same commutation relations which are satisfied by the matrices which have zeros everywhere except the position (i,j), where 1 stands. ( are sometimes called matrix units). Hence we conclude that the correspondence defines a representation of the Lie algebra in the vector space of polynomials of x_ij.